# Circulant vs normal

What is the relationship between the definition for a matrix to be circulant and to be normal? Does one imply the other?

Assume matrix $A$ is symmetric, then $A^T=A$ and clearly it is normal, but not circulant in general. However, if I assume that $A$ is circulant, looks like $A^TA=AA^T$, so is it normal?

(as I just found out on Wikipedia) Normal matrices are those matrices that are diagonalisable with respect to some othonormal basis for the standard (Hermitian) inner product of $\mathbb C^n$. And circulant matrices are those that are diagonalisable with respect to one particular basis, formed of vectors $\zeta^0,\zeta^1,\ldots,\zeta^{n-1}$ where $\zeta$ is an $n$-th root of unity (and runs through all such roots as one runs through the basis). This basis is othonormal for the standard inner product, so "circulant matrix" is a very special case of "normal matrix".